APPENDIX
Mathematical Notes
The references marked "Report" are to the writer's "Report on the Relativity Theory of Gravitation" for the Physical Society of London (Fleetway Press), where fuller mathematical details are given.
Probably the most complete treatise on the mathematical theory of the subject is H. Weyl's Raum, Zeit, Materie (Julius Springer, Berlin).
Note 1 (p. 20).
It is not possible to predict the contraction rigorously from the universally accepted electromagnetic equations, because these do not cover the whole ground. There must be other forces or conditions which govern the form and size of an electron; under electromagnetic forces alone it would expand indefinitely. The old electrodynamics is entirely vague as to these forces.
The theory of Larmor and Lorentz shows that if any system at rest in the aether is in equilibrium, a similar system in uniform motion through the aether, but with all lengths in the direction of motion diminished in FitzGerald's ratio, will also be in equilibrium so far as the differential equations of the electromagnetic field are concerned. There is thus a general theoretical agreement with the observed contraction, provided the boundary conditions at the surface of an electron behave in the same way. The latter suggestion is confirmed by experiments on isolated electrons in rapid motion (Kaufmann's experiment). It turns out that this requires an electron to suffer the same kind of contraction as a material rod; and thus, although the theory throws light on the adjustments involved in material contraction, it can scarcely be said to give an explanation of the occurrence of contraction generally.
